The main street is busy with "passiagata" in the evening after the hotel meals are over. Piazza del  Popolo is the centre point with Corso Matteotti one way and Corso Roma the other. Of course best in the day is Viale Verdi (starts as pedestrian)  the walk from the Piazza del Popolo along the park to the Terme Tettuccio. This itself is a 'must see' .

A good tip is to stop for a coffee at Cafe Biondi, corner of Viale Verdi and Piazza Del Popolo as there are good people watching opportunities there.

The most athmospheric and charming can be Via E Cavallotti. Start from Via Grocco just past the Tourist Board office. A nice cafe here on the corner and another typical one half way down by Via Palestro. The narrow street is full of smaller hotels and everyone sits outside and chats. Wonderful. Another era. Sometimes dance music wafts out of a hotel and you can see (mainly over 60's) having a great time. Via Cavallotti may not be the French Quarter or  Great Compton street but it has something. Do you know in the world of many streets like this that are not all cafes and bars? 

See Sports and activities section for info on  high quality places, Swimming Pools/Restaurant bars for younger people. La villa resort is excellent. Le panterie (.it) is another.